On limiting trace inequalities for vectorial differential operators

We establish that trace inequalities $$|D^{k-1}u|_{L^{frac{n-s}{n-1}}(mathbb{R}^{n},dmu)} leq c |mu|_{L^{1,n-s}(mathbb{R}^{n})}^{frac{n-1}{n-s}}|mathbb{A}[D]u|_{L^{1}(mathbb{R}^{n},dmathscr{L}^{n})}$$ hold for vector fields $uin C^{infty}(mathbb{R}^{n};mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $mathbb{A}[D]$ on $mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $mathbb{A}[D]$ are necessary. Here, $|mu|_{L^{1,lambda}}$ denotes the $(1,lambda)$-Morrey norm of the measure $mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($mathbb{A}$-)strict continuity of the associated trace operators on $text{BV}^{mathbb{A}}$.